To simplify [tex]\(\sqrt{-121}\)[/tex], let's follow the steps for handling the square root of a negative number.
1. Understanding the square root of a negative number:
- The square root of a negative number can be expressed in terms of the imaginary unit [tex]\(i\)[/tex], where [tex]\(i^2 = -1\)[/tex]. In particular, [tex]\(\sqrt{-1} = i\)[/tex].
2. Breaking down the expression:
- We need to simplify [tex]\(\sqrt{-121}\)[/tex]. Notice that [tex]\(\sqrt{-121}\)[/tex] can be rewritten as [tex]\(\sqrt{121 \cdot (-1)}\)[/tex].
3. Using the property of square roots:
- The property of square roots indicates that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]. Applying this to our expression, we get:
[tex]\[
\sqrt{-121} = \sqrt{121 \cdot (-1)} = \sqrt{121} \cdot \sqrt{-1}
\][/tex]
4. Simplifying each part:
- We know that [tex]\(\sqrt{121} = 11\)[/tex], because [tex]\(121\)[/tex] is a perfect square and [tex]\(11^2 = 121\)[/tex].
- And we have [tex]\(\sqrt{-1} = i\)[/tex], as defined by the imaginary unit.
5. Combining the results:
- Multiplying these results together, we get:
[tex]\[
\sqrt{121} \cdot \sqrt{-1} = 11 \cdot i
\][/tex]
6. Final simplified expression:
- Therefore, [tex]\(\sqrt{-121}\)[/tex] simplifies to [tex]\(11i\)[/tex].
Hence, the correct simplified form of [tex]\(\sqrt{-121}\)[/tex] is [tex]\(11i\)[/tex].
The correct answer is:
[tex]\[ \boxed{11i} \][/tex]
